Geometric Invariant Theory and Generalized Eigenvalue Problem II

Abstract

Let G be a connected reductive subgroup of a complex connected reductive group Ĝ. Fix maximal tori and Borel subgroups of G and Ĝ. Consider the cone LR • ( Ĝ, G) generated by the pairs (ν, ν) of strictly dominant characters such that V ν is a submodule of V ν . The main result of this article is a bijective parametrisation of the faces of LR • ( Ĝ, G). We also explain when such a face is contained in another one.In way, we obtain results about the faces of the Dolgachev-Hu's Gample cone. We also apply our results to reprove known results about the moment polytopes.

Inequalities for moment cones of finite-dimensional representations

“…Here, Ω denotes the set of weights of M and R G,− denotes the set of negative roots of G. By applying theorem 1.1 to the lower-dimensional scenario, the Horn condition can be explicitly stated as a set of linear inequalities that have to be satisfied by κ H . Tangent maps and their determinants have been studied in great generality by Ressayre and Belkale from an algebro-geometric point of view [25,29,30], and our theorem 1.1 and proposition 1.2 can also be deduced from their results. In these works, the non-vanishing of the determinant has been in turn been translated into a cohomological condition.…”

The combinatorics of quiver representations

“…Other results such as Theorem 2.9 or elements in Section 3 were obtained in [17]. Results about some small faces were obtain in [31,29]. A new proof of Theorem 5.1 can be found in [30].…”

GIT-cones and quivers

“…This point of view was used in [1,2,13,16]. While the GIT and the quiver approaches to the Horn problem have traditionally been distinct, we hope to show that the GIT approach is more universal when studying (Q, β).…”